Theorems · Theorem · general topology
Set.Countable.isPathConnected_compl_of_one_lt_rank
∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] [inst_2 : TopologicalSpace E] [ContinuousAdd E]
[ContinuousSMul ℝ E], 1 < Module.rank ℝ E → ∀ {s : Set E}, s.Countable → IsPathConnected sᶜIn a real vector space of dimension > 1, the complement of any countable set is path
connected.
- Defined in
- Mathlib.Analysis.Normed.Module.Connected
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 170 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites65
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Setstatement and proof · cited by 53,352
- Realstatement and proof · cited by 25,697
- TopologicalSpacestatement and proof · cited by 24,529
- Modulestatement and proof · cited by 20,661
- Semiringproof · cited by 13,802
- AddCommGroupstatement and proof · cited by 12,871
- AddCommMonoidproof · cited by 12,281
- PartialOrderproof · cited by 6,410
- Set.ofPredproof · cited by 6,101
- Algebra.algebraMapproof · cited by 4,706
- Compl.complstatement and proof · cited by 2,925
Cited by2
Results whose statement or proof uses this declaration.
- isPathConnected_compl_singleton_of_one_lt_rankproof · cited by 3
- Set.Countable.isConnected_compl_of_one_lt_rankproof · cited by 1